Recently, various generalizations of left braces were introduced to study set-theoretic solutions that are not necessarily involutive. In [1]}, Guarnieri and Vendramin introduced skew left braces to study bijective non-degenerate solutions. As one can expect, skew left braces give structural informations on bijective non-degenerate solutions (see, for example, [2]}, [3]}, [1]}, ). Skew left braces were furtherly generalized by Catino, Colazzo and Stefanelli in [5]} to deal with left non-generated solutions, not necessarily bijective. Technically, they introduced the notion of left cancellative left semi-brace, that is a triple \((B,+,\circ )\)  such that \((B,+)\)  is a left cancellative semigroup, \((B,\circ )\)  is a group and the equality
\(a\circ (b+c)=a\circ b+a\circ (a^-+c)\) 
